Article Contents
Article Contents

# Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity

• We will focus on the existence and concentration of nodal solutions to the following critical nonlinear Schrödinger equations in $\Bbb R^2$ $$-\epsilon^2\triangle u_{\epsilon}+V(x)u_{\epsilon}=K(x) |u_{\epsilon}|^{p-2}u_{\epsilon}e^{\alpha_{0}|u_{\epsilon}| ^{2}},\quad u_{\epsilon}\in H^1(\Bbb R^2),$$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, and $\epsilon>0$ is a small constant. For the positive potential $V(x)$ which decays at infinity like $(1+|x|)^{-\alpha}$ with $0 < \alpha \le 2$, we will show that a nodal solution with one positive and one negative peaks exists, and concentrates around local minimum points of the related ground energy function $G(\xi)$ of the Schrödinger equation $-\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{2}}$.
Mathematics Subject Classification: Primary: 35B33, 35J60; Secondary: 35Q55.

 Citation:

•  [1] C. O. Alves and S. H. M. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl., 296 (2004), 563-577.doi: 10.1016/j.jmaa.2004.04.022. [2] C. O. Alves and S. H. M. Soares, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations, 234 (2007), 464-484.doi: 10.1016/j.jde.2006.12.006. [3] A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.doi: 10.4171/JEMS/24. [4] A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332. [5] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18.doi: 10.1007/BF02787822. [6] T. Bartsch, C. Mónica and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation, Math. Ann., 338 (2007), 147-185.doi: 10.1007/s00208-006-0071-1. [7] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345.doi: 10.1007/BF00250555. [8] J. Byeon and Z.-Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. Eur. Math. Soc., 8 (2006), 217-228.doi: 10.4171/JEMS/48. [9] D. Cao, Nontrivial solutions of semilinear elliptic equation with critical exponent in $\mathbbR^{2}$, Comm. Partial Differential Equations, 17 (1992), 407-435.doi: 10.1080/03605309208820848. [10] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.doi: 10.1007/BF01189950. [11] J. M. do Ó and M. A. S. Souto, On a class of nonlinear Schrödinger equations in $\mathbbR^{2}$ involving critical growth, J. Differential Equations, 174 (2001), 289-311.doi: 10.1006/jdeq.2000.3946. [12] M. Fei and H. Yin, Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials, Pacific. J. Math., 244 (2010), 261-296.doi: 10.2140/pjm.2010.244.261. [13] D. G. de Figueiredo, J. M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure. Appl. Math., 55 (2002), 135-152.doi: 10.1002/cpa.10015. [14] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbbR^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.doi: 10.1007/BF01205003. [15] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin-New York, 1998. [17] Z. Liu and Z.-Q. Wang, Sign-changing solutions of nonlinear elliptic equations, Front. Math. China, 3 (2008), 221-238.doi: 10.1007/s11464-008-0014-0. [18] E. S. Noussair and J. Wei, On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Indiana Univ. Math. J., 46 (1997), 1255-1271.doi: 10.1512/iumj.1997.46.1401. [19] E. S. Noussair and J. Wei, On the location of spikes and profile of nodal solutions for a singularly perturbed Neumann problem, Comm. Partial Differential Equations, 23 (1998), 793-816.doi: 10.1080/03605309808821366. [20] Y. Sato, Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency, Commun. Pure. Appl. Anal., 7 (2008), 883-903.doi: 10.3934/cpaa.2008.7.883. [21] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.doi: 10.1007/BF02096642. [22] X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.doi: 10.1137/S0036141095290240. [23] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996.doi: 10.1007/978-1-4612-4146-1. [24] H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647.doi: 10.1016/j.jde.2009.03.002.